Hall Effect of Light

We derive the semiclassical equation of motion for the wave packet of light taking into account the Berry curvature in momentum-space. This equation naturally describes the interplay between orbital and spin angular momenta, i.e., the conservation of the total angular momentum of light. This leads to the shift of wave-packet motion perpendicular to the gradient of the dielectric constant, i.e., the polarization-dependent Hall effect of light. An enhancement of this effect in photonic crystals is also proposed.

Figure 1

Transverse shift of light beams in the refraction and reflection at an interface.

Figure 2

Transverse shift in the center of the transmitted and reflected wave packets of light for (a) $n=2.0$ and (b) $n=0.8$ with the incident wave packet being right-circularly polarized. The solid and dashed lines for the analytic results Eq. (4). The solid circles and squares are the results of simulation. ${\lambda }_c^I=2\pi /k_c^I$ is the wavelength of incident wave packet. $l_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}$ is an arbitrary length scale. The sign of the shifts is reversed for the left-circularly polarized light.

Figure 3

(a) Dielectric function and (b) band structure of the 2D photonic crystal. The Brillouin zone is shown in Figs. 4, 5b.

Figure 4

Example of the BC of the 2D photonic crystal. It is prominent at the corners of the Brillouin zone.

Figure 5

(a) Real-space trajectory of the center of the wave packet. (b) Momentum-space trajectory of the center of the wave packet. The arrows in the same colors in (a) and (b) refer to the same photonic modes.